Since \(\mu \sim O(1)\)  and \(\lambda \sim O(1)\) , \(K(k,l)\)  is of order \(1/N^2\) , which is consistent with the switchback effect of circuit complexity [1]}. In fact, we will show that to make sure \(K(k,l)\sim 1/N^2\) , the complexity metric Eq. (REF ) is the only appropriate metric up to constant corrections. Consider typical sectional curvatures for the general metric Eq. (REF ), if the asymptotic behavior of the penalty is \(f(k)\sim O(k^{\gamma })\) , then Eq. (REF ) gives the asymptotic behavior of curvatures,
\(K(k,l)\sim O(N^{-\gamma }).\) 
